3
Rational Functions A rational function can have more than one vertical asymptote, but it can have at most one horizontal asymptote. A rational function f ( x ) is a function that can be written as where p ( x ) and q ( x ) are polynomial functions and q ( x ) 0.

4
Vertical Asymptotes If p ( x ) and q ( x ) have no common factors, then f ( x ) has vertical asymptote(s) when q ( x ) = 0. Thus the graph has vertical asymptotes at the zeros of the denominator.

5
Since the zeros are 1 and -1. Thus the vertical asymptotes are x = 1 and x = -1. Vertical Asymptotes Find the vertical asymptote of Example: V.A. is x = a, where a represents real zeros of q ( x ).

6
Horizontal Asymptotes The horizontal asymptote is determined by looking at the degrees of p ( x ) and q ( x ). A rational function f ( x ) is a function that can be written as where p ( x ) and q ( x ) are polynomial functions and q ( x ) 0.

7
Horizontal Asymptotes a.If the degree of p ( x ) is less than the degree of q ( x ), then the horizontal asymptote is y = 0. b. If the degree of p ( x ) is equal to the degree of q ( x ), then the horizontal asymptote is c. If the degree of p ( x ) is greater than the degree of q ( x ), then there is no horizontal asymptote.